The Law of Sines & Cosines — Drills
The Law of Sines & Cosines has 6 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Area — Space inside, measured in square units.
- Included angle — The angle between two named sides. The A in SAS must be included.
- Sine — sin θ = oppositehypotenuse.
- Law of sines — asin A = bsin B = csin C.
- Law of cosines — c2 = a2 + b2 - 2abcos C.
Third
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
In triangle ABC, A = 55° and B = 62°. Find C.
Worked answer
63
180-55-62 = 63
Problem 2
In triangle ABC, A = 68° and B = 54°. Find C.
Worked answer
58
180-68-54 = 58
Sines
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
In triangle ABC, A = 49°, a = 19 and B = 55°. Find b to one decimal place.
Worked answer
20.6
b = 19sin 55°sin 49°
≈ 20.6
Problem 2
In triangle ABC, A = 60°, a = 11 and B = 48°. Find b to one decimal place.
Worked answer
9.4
b = 11sin 48°sin 60°
≈ 9.4
Cosines
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A triangle has sides 11 and 8 with an included angle of 45°. Find the third side to one decimal place.
Worked answer
7.8
121+64-2(11)(8)cos 45°
c ≈ 7.8
Problem 2
A triangle has sides 3 and 7 with an included angle of 120°. Find the third side to one decimal place.
Worked answer
8.9
9+49-2(3)(7)cos 120°
c ≈ 8.9
Angle
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A triangle has sides 7, 8 and 12. Find the largest angle to the nearest degree.
Worked answer
106
144 = 49+64-2(7)(8)cos C
cos C = -0.277
C ≈ 106°
Problem 2
A triangle has sides 5, 6 and 10. Find the largest angle to the nearest degree.
Worked answer
131
100 = 25+36-2(5)(6)cos C
cos C = -0.650
C ≈ 131°
Area
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Find the area of a triangle with sides 13 and 11 and an included angle of 30°.
Worked answer
35.75
12(13)(11)sin 30° = 35.75
Problem 2
Find the area of a triangle with sides 5 and 10 and an included angle of 150°.
Worked answer
12.5
12(5)(10)sin 150° = 12.5
Inequalities
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Can a triangle have sides 7, 5 and 6?
- Yes
- No
- Only if right-angled
- Only if obtuse
Worked answer
Yes
7+5 = 12, which exceeds 6
So the sides close up
Problem 2
Can a triangle have sides 2, 4 and 7?
- Yes
- No
- Only if right-angled
- Only if obtuse
Worked answer
No
2+4 = 6, which does not reach 7
So the short sides cannot meet